#273246

order	512
nilpotency class	3
isoclinism	isoclinic class of #273243
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

898 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,11,12,8)(4,14,15,7)(5,9)(6,10)(13,16)(17,18,22,26)(19,27,29,32)(20,30,31,24)(21,28,23,25);(1,3)(2,7)(4,13)(5,15)(6,12)(8,9)(10,14)(11,16)(17,19)(18,20)(21,30)(22,29)(23,24)(25,32)(26,31)(27,28);(3,12)(5,13)(7,14)(9,16)(17,20)(18,24)(19,28)(21,27)(22,31)(23,32)(25,29)(26,30);(1,4,6,15)(2,8,10,11)(3,13,12,5)(7,9,14,16)(17,20)(18,19)(21,32)(22,31)(23,27)(24,28)(25,30)(26,29)

Polycyclic presentation:

2^9|a^2=f,d^2=hi,e^2=i,g^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=gh,e^c=ehi,g^c=gh,e^d=ei,f^d=fhi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180442);